Scheduling jobs with truncated exponential learning functions

In this paper we consider the single machine scheduling problem with truncated exponential learning functions. By the truncated exponential learning functions, we mean that the actual job processing time is a function which depends not only on the total normal processing times of the jobs already processed but also on a control parameter. The use of the truncated function is to model the phenomenon that the learning of a human activity is limited. We show that even with the introduction of the proposed model to job processing times, several single machine problems remain polynomially solvable. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without truncated exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms.     

Worst-case analysis for flow shop scheduling problems with an exponential learning effect

A real industrial production phenomenon, referred to as learning effects, has drawn increasing attention. However, most research on this issue considers only single machine problems. Motivated by this limitation, this paper considers flow shop scheduling problems with an exponential learning effect. By the exponential learning effect, we mean that the processing time of a job is defined by an exponent function of its position in a processing permutation. The objective is to minimize one of the four regular performance criteria, namely, the total completion time, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present heuristic algorithms by using the optimal permutations for the corresponding single-machine scheduling problems. We also analyse the worst-case bound of our heuristic algorithms.     

Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect

In this paper we consider the single machine scheduling problems with exponential sum-of-logarithm-processing-times based learning effect. By the exponential sum-of-logarithm-processing-times based learning effect, we mean that the processing time of a job is defined by an exponent function of the sum of the logarithm of the processing times of the jobs already processed. We consider the following objective functions: the makespan, the total completion time, the sum of the quadratic job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single-machine scheduling problems with time and position dependent processing times

We consider single-machine scheduling problems with time and position dependent job processing times. In many industrial settings, the processing time of a job changes due to either job deterioration over time or machine/worker’s learning through experiences. In the models we study, each job has its normal processing time. However, a job’s actual processing time depends on when its processing starts and how many jobs have completed before its start. We prove that the classical SPT (Shortest Processing Time) rule remains optimal when we minimize the makespan or the total completion time. For problems of minimizing the total weighted completion time, the maximum lateness, and the discounted total weighted completion time, we present heuristic sequencing rules and analyze the worst-case bounds for performance ratios. We also show that these heuristic rules can be optimal under some agreeable conditions between the normal processing times and job due dates or weights.     

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ? 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Worst-case behavior of simple sequencing rules in flow shop scheduling with general position-dependent learning effects

A real industrial production phenomenon, referred to as learning effects, has drawn increasing attention. However, most research on this issue considers only single machine problems. Motivated by this limitation, this paper considers flow shop scheduling problems with a general position-dependent learning effects. By the general position-dependent learning effects, we mean that the actual processing time of a job is defined by a general non-increasing function of its scheduled position. The objective is to minimize one of the five regular performance criteria, namely, the total completion time, the makespan, the total weighted completion time, the total weighted discounted completion time, and the sum of the quadratic job completion times. We present heuristic algorithms by using the optimal permutations for the corresponding single machine scheduling problems. We also analyze the worst-case bound of our heuristic algorithms.     

Single-machine scheduling problems with an actual time-dependent deterioration

In this paper, we consider the single machine scheduling problems with an actual time-dependent deterioration effect. By the actual time-dependent deterioration effect, we mean that the processing time of a job is defined by increasing function of total actual processing time of jobs in front of it in the sequence. We show that even with the introduction of an actual time-dependent deterioration to job processing times, makespan minimization problem, total completion time minimization problem, the total lateness, and the sum of the quadratic job completion times minimization problem remain polynomially solvable, respectively. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, and the total tardiness minimization problem can be solved in polynomial time under certain conditions.     

Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

Single machine past-sequence-dependent delivery times scheduling with general position-dependent and time-dependent learning effects

This paper studies the single machine past-sequence-dependent (p-s-d) delivery times scheduling with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. We consider the following objective functions: the makespan, the total completion time, the sum of the θ$\theta$th (θ?0$\theta ?0$) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of minimization of the makespan, the total completion time, the sum of the θ$\theta$th (θ?0$\theta ?0$) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, the maximum tardiness minimization problem and the total tardiness minimization problem can be solved in polynomial time under certain conditions.     

Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions

Scheduling with learning effects has received growing attention nowadays. A well-known learning model is called ‘position-based learning’ in which the actual processing time of a job is a non-increasing function of its position to be processed. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases. Motivated by this observation, we propose two truncated learning models in single-machine scheduling problems and two-machine flowshop scheduling problems with ordered job processing times, respectively, where the actual processing time of a job is a function of its position and a control parameter. Under the proposed learning models, we show that some scheduling problems can be solved in polynomial time. In addition, we further analyse the worst-case error bounds for the problems to minimize the total weighted completion time, discounted total weighted completion time and maximum lateness.     

Single-machine scheduling with a sum-of-actual-processing-time-based learning effect

In this paper we consider the single-machine scheduling problems with a sum-of-actual-processing-time-based learning effect. By the sum-of-actual-processing-time-based learning effect, we mean that the processing time of a job is defined by a function of the sum of the actual processing time of the already processed jobs. We show that even with the introduction of the sum-of-actual-processing-time-based learning effect to job processing times, the makespan minimization problem, the total completion time minimization problem, the total completion time square minimization problem, and some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem remain polynomially solvable, respectively.     

Scheduling problems with past-sequence-dependent setup times and general effects of deterioration and learning

In this paper we consider the single-machine setup times scheduling with general effects of deterioration and learning. By the general effects of deterioration and learning, we mean that the actual job processing time is a general function of the processing times of the jobs already processed and its scheduled position. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We show that the problems to minimize the makespan, the sum of the δ$\delta$th (δ>0$\delta >0$) power of job completion times, the total lateness are polynomially solvable. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness (tardiness) minimization problem, the total tardiness minimization problem can be solved in polynomial time under certain conditions.     

Scheduling jobs with position-dependent processing times

The paper is devoted to some single machine scheduling problems, where job processing times are defined by functions dependent on their positions in the sequence. It is assumed that each job is available for processing at its ready time. We prove some properties of the special cases of the problems for the following optimization criteria: makespan, total completion time and total weighted completion time. We prove strong NP-hardness of the makespan minimization problem for two different models of job processing time. The reductions are done from the well-known 3-Partition Problem. In order to solve the makespan minimization problems, we suggest the Earliest Ready Date algorithms, for which the worst-case ratios are calculated. We also prove that the makespan minimization problem with job ready times is equivalent to the maximum lateness minimization problem.     

Some single-machine scheduling problems with actual time and position dependent learning effects

In this paper we study some single-machine scheduling problems with learning effects where the actual processing time of a job serves as a function of the total actual processing times of the jobs already processed and of its scheduled position. We show by examples that the optimal schedules for the classical version of problems are not optimal under this actual time and position dependent learning effect model for the following objectives: makespan, sum of kth power of the completion times, total weighted completion times, maximum lateness and number of tardy jobs. But under certain conditions, we show that the shortest processing time (SPT) rule, the weighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and the modified Moore’s Algorithm can also construct an optimal schedule for the problem of minimizing these objective functions, respectively.     

Single-machine scheduling with effects of exponential learning and general deterioration

The purpose of this study is to explore the single-machine scheduling with the effects of exponential learning and general deterioration. By the effects of exponential learning and general deterioration, we meant that job processing time is decided by the functions of their starting time and positions in the sequence. Results showed that with the introduction of learning effect and deteriorating jobs to job processing time, single-machine makespan, and sum of completion time (square) minimization problems remained polynomially solvable, respectively. But for the following objective functions: the weighted sum of completion time and the maximum lateness, this paper proved that the weighted smallest basic processing time first (WSPT) rule and the earliest due date first (EDD) rule constructed the optimal sequence under some special cases, respectively.     

Research on permutation flow shop scheduling problems with general position-dependent learning effects

Machine learning exists in many realistic scheduling situations. This study focuses on permutation flow shop scheduling problems, where the actual processing time of a job is defined by a general non-increasing function of its scheduled position, i.e., general position-dependent learning effects. The objective functions are to minimize the total completion time, the makespan, the total weighted completion time, and the total weighted discounted completion time, respectively. To solve these problems, we present approximation algorithms based on the optimal permutations for the corresponding single machine scheduling problems and analyze their worst-case error bound.     

分类数很大时的成组排序问题

成组排序具有深刻的实际应用背景,是近年来国外研究得较多的一个热点.已有的某些动态规划算法的复杂性随分类数的增长呈指数型增长趋势,本文用“归并”和解不超过四个新的子问题的方法把分类数较大时的问题转化为分类数较小时的相应问题,简化了问题的求解.     

Single-machine scheduling problems with actual time-dependent and job-dependent learning effect

In this study, we introduce an actual time-dependent and job-dependent learning effect into single-machine scheduling problems. We show that the complexity results of the makespan minimization problem and the sum of weighted completion time minimization problem are all NP-hard. The complexity result of the maximum lateness minimization problem is NP-hard in the strong sense. We also provide three special cases which can be solved by polynomial time algorithms.     

Machine scheduling with a rate-modifying activity

Motivated by a problem commonly found in electronic assembly lines, this paper deals with the problem of scheduling jobs and a rate-modifying activity on a single machine. A rate-modifying activity is an activity that changes the production rate of the equipment under consideration. Hence the processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. The decisions under consideration are when to schedule the rate-modifying activity and the sequence of jobs to optimize some performance measure.In this paper, we develop polynomial algorithms for solving problems of minimizing makespan, and total completion time respectively. We also develop pseudo-polynomial algorithms for solving problems of total weighted completion time under the agreeable ratio assumption. We prove that the problem of minimizing maximum lateness is NP-hard and also provide a pseudo-polynomial time algorithm to solve it optimally.     

Single machine scheduling with general time-dependent deterioration, position-dependent learning and past-sequence-dependent setup times

The paper deals with single machine scheduling problems with setup time considerations where the actual processing time of a job is not only a non-decreasing function of the total normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We consider the following objective functions: the makespan, the total completion time, the sum of the δth (δ ≥ 0) power of job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the δ th (δ ≥ 0) power of job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.